Questions tagged [hardy-spaces]
For questions about Hardy spaces. Use the other related tag like (tag: complex-analysis) or (operator-theory).
212 questions
-2 votes
0 answers
83 views
Disk version of an inner–outer claim: when does (|f|=1) a.e. force no inner factor?
Let $f$ be a function of bounded type (Nevanlinna class) in the unit disk $\mathbb{D}$. Suppose: $f$ is analytic and zero-free in $\mathbb{D}$; Both $f$ and $1/f$ are of bounded type in $\mathbb{D}$; ...
0 votes
0 answers
48 views
Understanding the orthogonal projection of functions in $L^2([-\pi,\pi])$
I am reading Stein and Shakarchi's Real Analysis, and have got stuck on an example of how orthogonal projections work. The book states the following: Consider $L^2([-\pi,\pi])$, and let $S$ denote ...
- 355
0 votes
0 answers
40 views
Equivalence of the ${\Lambda_{a}(\mathbb{R}^n)}$ space
This is from the book Hardy spaces on the Euclidean space by Akihito Uchiyama. In page 14-16, the author give the following lemma: Lemma 1.2. Let $a>0$, $\varphi \in \mathcal{D}$, $\int\varphi(x)dx ...
- 234
2 votes
0 answers
139 views
When is this operator bounded on Hardy space?
Let $ 1 \leq p \leq +\infty $ and $ g \in \mathcal{H}(\mathbb{D}) $. Consider the multiplication operator $ M_g(f) = g f $. I want to characterize the functions $ g $ for which the operator $ M_g : H^...
- 53
1 vote
0 answers
34 views
Where is an outer function in $H^{\infty}$ analytic?
Let $H^{\infty}(\mathbb{D})$ denote the Hardy space of the unit disc consisting of the bounded analytic functions on the open unit disk $\mathbb{D}$. An outer function $f \in H^{\infty}(\mathbb{D})$ ...
- 134
2 votes
0 answers
55 views
Is atomic Hardy space $H_1(\mathbb{R})$ separable?
The atomic Hardy space $H_1(\mathbb{R})$ is space of functions in $L_1(\mathbb{R})$ of the from $$ f(x) = \sum_{i=0}^\infty \lambda_i a_i $$ where $(\lambda_i)_i$ is in $\ell_1$, and each $a_i$ is an ...
- 91
1 vote
1 answer
63 views
Proving $A:=\{x\in C(\bar{D}):x$ is analytic on $D\}$ a Banach Space
I was solving some exercises on proving Banach Space. A question is given in the following way: Let $D$ denote the open unit disk in $\mathbf{C}$. Let $A:=\{x\in C(\bar{D}):x$ is analytic on $D\}$. ...
- 334
2 votes
0 answers
88 views
Boundary behaviour of non-constant inner functions
Can a non-constant inner function map the unit circle on a proper subset of itself? Intuitively, I guess the answer is negative and I suppose there should be a relatively simple argument but I could ...
- 415